Questions: Crystal Nguyen Question 1, 7.2.1 Part 3 of 3 HW Score: 3.75%, 0.75 of 20 points Points: 0.75 of 1 Save Refer to the accompanying data display that results from a simple random sample of times (minutes) between eruptions of the Old Faithful geyser. The confidence level of 95% was used. Complete parts (a) and (b) below. TInterval (85.74,91.76) x̄=88.75 Sx=8.897431411 n=36 a. Express the confidence interval in the format that uses the "less than" symbol. Round the confidence interval limits given that the original times are all rounded to one decimal place. 85.74 min < μ < 91.76 min (Round to two decimal places as needed.) b. Identify the best point estimate of μ and the margin of error. The point estimate of μ is 88.75 minutes. (Round to two decimal places as needed.) The margin of error is E= minutes. (Round to two decimal places as needed.)

Crystal Nguyen
Question 1, 7.2.1
Part 3 of 3
HW Score: 3.75%, 0.75 of 20 points
Points: 0.75 of 1
Save

Refer to the accompanying data display that results from a simple random sample of times (minutes) between eruptions of the Old Faithful geyser. The confidence level of 95% was used. Complete parts (a) and (b) below.

TInterval
(85.74,91.76)
x̄=88.75
Sx=8.897431411
n=36

a. Express the confidence interval in the format that uses the "less than" symbol. Round the confidence interval limits given that the original times are all rounded to one decimal place.

85.74 min < μ < 91.76 min
(Round to two decimal places as needed.)

b. Identify the best point estimate of μ and the margin of error.

The point estimate of μ is 88.75 minutes.
(Round to two decimal places as needed.)
The margin of error is E= minutes.
(Round to two decimal places as needed.)

Transcript text: Crystal Nguyen Question 1, 7.2.1 Part 3 of 3 HW Score: $3.75 \%, 0.75$ of 20 points Points: 0.75 of 1 Save Refer to the accompanying data display that results from a simple random sample of times (minutes) between eruptions of the Old Faithful geyser. The confidence level of $95 \%$ was used. Complete parts (a) and (b) below. \[ \begin{array}{l} \quad \text { TInterval } \\ (85.74,91.76) \\ \bar{x}=88.75 \\ S x=8.897431411 \\ n=36 \end{array} \] a. Express the confidence interval in the format that uses the "less than" symbol. Round the confidence interval limits given that the original times are all rounded to one decimal place. \[ 85.74 \min <\mu<91.76 \text { min } \] (Round to two decimal places as needed.) b. Identify the best point estimate of $\mu$ and the margin of error. The point estimate of $\mu$ is 88.75 minutes. (Round to two decimal places as needed.) The margin of error is $E=$ $\square$ minutes. (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Confidence Interval Expression

The confidence interval for the mean eruption time of the Old Faithful geyser is expressed as: \[ 85.74 \, \text{min} < \mu < 91.76 \, \text{min} \]

Step 2: Point Estimate of $ \mu $

The best point estimate of the population mean $ \mu $ is given by the sample mean: \[ \bar{x} = 88.75 \, \text{min} \]

Step 3: Margin of Error Calculation

To calculate the margin of error $ E $, we use the formula: \[ E = \frac{Z \times \sigma}{\sqrt{n}} \] where:

$ Z = 1.96 $ (Z-score for a 95% confidence level)
$ \sigma = 8.897431411 $ (sample standard deviation)
$ n = 36 $ (sample size)

Substituting the values: \[ E = \frac{1.96 \times 8.897431411}{\sqrt{36}} = 2.91 \, \text{min} \]

Final Answer

The results are summarized as follows:

The confidence interval is $ 85.74 \, \text{min} < \mu < 91.76 \, \text{min} $.
The point estimate of $ \mu $ is $ 88.75 \, \text{min} $.
The margin of error is $ E = 2.91 \, \text{min} $.

Thus, the final boxed answers are: \[ \boxed{85.74 \, \text{min} < \mu < 91.76 \, \text{min}} \] \[ \boxed{\bar{x} = 88.75 \, \text{min}} \] \[ \boxed{E = 2.91 \, \text{min}} \]

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